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We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs)as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They areobtained by gauging $G$-equivariant invertible spin-TQFTs, or, in physicslanguage, gauging the interacting fermionic Symmetry Protected Topologicalstates (SPTs) with a finite group $G$ symmetry. We use the fact that the latterare classified by Pontryagin duals to spin-bordism groups of the classifyingspace $BG$. We also consider unoriented analogues, that is $G$-equivariantinvertible pin$^\pm$-TQFTs (fermionic time-reversal-SPTs) and their gauging. Wecompute these groups for various examples of abelian $G$ using Adams spectralsequence and describe all corresponding TQFTs via certain bordism invariants indimensions 3, 4, and other. This gives explicit formulas for the partitionfunctions of spin-TQFTs on closed manifolds with possible extended operatorsinserted. The results also provide explicit classification of 't Hooftanomalies of fermionic QFTs with finite abelian group symmetries in onedimension lower. We construct new anomalous boundary deconfined spin-TQFTs(surface fermionic topological orders). We explore SPT and SET (symmetryenriched topologically ordered) states, and crystalline SPTs protected byspace-group (e.g. translation $\mathbb{Z}$) or point-group (e.g. reflection,inversion or rotation $C_m$) symmetries, via the layer-stacking construction.

Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms